What is the difference between VoIP and VoLTE?

VoIP refers to Voice Over IP, which is a methodology of delivering the voice communication data over the internet. Voice over IP (VoIP) refers to sending voice data over the internet. Initially this was only possible using computers (connected or WiFi). An example would be Skype. 

With mobile phones becoming popular (pre-smartphone mostly but also some smartphones) there was an attempt to port VoIP over mobile phones and this was generally referred to as mobile VoIP (mVoIP). You could use cellular data connectivity or WiFi for VoIP in this case. Nowadays, VoIP is referred for Over The Top (OTT) applications that may use PC's, tablets, smartphones, etc. Example would be Skype, Viber, WhatsApp, etc.

Packets are packets and the only thing that knows they are voice is the application that receives than & processes them eg your Skype app. As such that is Layer 7 and they go through L1-L6 exactly as every other packet. They could be going over LTE, over WiFi or over ADSL - it doesn't matter = packets are packets.

The cellular world used to have 2 separate core networks called CS (circuit switched) that was used for voice and PS (packet switched) for data. To simplify the network, they started moving Voice over to PS part. For 3G/HSPA this became known as VoHSPA (Voice over HSPA) and for 4G/LTE it is known as VoLTE (Voice over LTE). Since HSPA/LTE is data only network, the networks realised that they can use WiFi for the access network in the same way. Hence VoWiFi is using WiFi for access but the PS network for guaranteeing the quality of service (QoS). 

The Voice over LTE, VoLTE scheme was devised as a result of operators seeking a standardised system for transferring traffic for voice over LTE. Originally LTE was seen as a completely IP cellular system just for carrying data, and operators would be able to carry voice either by reverting to 2G / 3G systems or by using VoIP in one form or another.

VoLTE is a specific type of VoIP designed into the LTE standard, so it has lots of very specific optimisations and accelerations.  In effect, the packets are treated as "special" by the whole LTE network. 

That means it will work better (lower latency, less power, etc) - and as you say, be able to handover to other LTE cells or to 3G/2G legacy networks.

Signals Research Group (very good, very respected consultancy) & Spirent did some testing and confirmed this.

Commercially deployed Voice-over-LTE (VoLTE) technology provides better service than 3G Circuit Switched (CS) voice and Internet calling services and Skype software.
The analysis found that VoLTE users enjoyed considerably better call quality, faster connectivity and substantially less drain on the batteries of their mobile devices. 

The report evaluates several attributes including call setup time, call reliability, call quality, network resource requirements, and the impact on battery life. The report’s highlights include:

• VoLTE call quality greatly exceeded that of 3G CS voice and was measurable higher than the HD voice service offered by Skype
• With network loading, and in particular with background applications running on the mobile phone and transferring data with the network, the VoLTE results were considerably better than Skype
• VoLTE call setup time was nearly twice as fast as 3G Circuit Switched Fallback (CSFB) call setup
• VoLTE used substantially fewer network resources than Skype voice, which in turn resulted in longer estimated device battery life for the subscriber and a more efficient network for the operator
• When leaving LTE coverage, VoLTE calls were successfully handed over to 3G CS voice. The network’s enhanced Single Radio Voice Call Continuity ensures calls continue without interruption

“The clear choice for mobile network operators should be a no brainer,” said Emil Olbrich, vice president of Network Technologies for SRG. “VoLTE delivers a consistently higher call quality than OTT voice applications. Operators should also strongly consider the long-term network savings associated with VoLTE, given that it takes advantage of advanced features that allow it to use far less network resources when it comes to supporting a voice call when compared with current OTT voice applications.” 

So how do you compare VoLTE, VoWiFi and VoIP. I found this interesting table

As can be seen in the table, VoLTE should provide you the best quality, followed by VoWiFi (SIP VoIP) and VoIP would be the worst. Again, the worst may still be good enough for most people most of the time.